This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A377974 #19 Aug 03 2025 10:32:45
%S A377974 1,0,-30,-540,-867660,-31107300,-33668157900,-1795572812400,
%T A377974 -1477793386682970,-103845834995498100,-69550699526934273180,
%U A377974 -6017200267937951322660,-3426636160378174348594500,-349303370036461528632524580,-174458882971934188146144343320,-20314204536496741742949242177040
%N A377974 Expansion of the 1920th root of the series 2*E_4(x) - E_8(x), where E_4 and E_8 are the Eisenstein series of weight 4 and weight 8.
%C A377974 Let R = 1 + x*Z[[x]] denote the set of integer power series with constant term equal to 1. Let P(n) = {g^n, g in R}. The Eisenstein series E_4(x) lies in P(8) (Heninger et al.). Since E_8(x) = E_4(x)^2, it follows that E_8(x) lies in P(16).
%C A377974 We claim that the series 2*E_4(x) - E_8(x) belongs to P(1920).
%C A377974 Proof.
%C A377974 E_4(x) = 1 + 240*Sum_{n >= 1} sigma_3(n)*x^n. Hence,
%C A377974 2*E_4(x) - E_8(x) = 2*E_4(x) - E_4(x)^2 = 1 - 240^2*( Sum_{n >= 1} sigma_3(n) )^2 is in the set R.
%C A377974 Hence, 2*E_4(x) - E_8(x) == 1 mod(240^2) == 1 (mod (2^8)*(3^2)*(5^2)).
%C A377974 It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_4(x) - E_8(x) belongs to P((2^7)*3*5) = P(1920). End Proof.
%H A377974 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%F A377974 a(n) ~ c / (r^n * n^(1921/1920)), where r = 0.004019427095115250686492968205049012182922598389629390919504184161606551652... is the root of the equation Sum_{k>=1} sigma_3(k) * r^k = 1/240 and c = -0.00052087420429807426289253718287... - _Vaclav Kotesovec_, Aug 03 2025
%p A377974 with(numtheory):
%p A377974 E := proc (k) local n, t1; t1 := 1 - 2*k*add(sigma[k-1](n)*q^n, n = 1..30)/bernoulli(k); series(t1, q, 30) end:
%p A377974 seq(coeftayl((2*E(4) - E(8))^(1/1920), q = 0, n),n = 0..20);
%t A377974 terms = 20; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E4[x] - E8[x])^(1/1920), {x, 0, terms}], x] (* _Vaclav Kotesovec_, Aug 03 2025 *)
%Y A377974 Cf. A004009 (E_4), A008410 (E_8), A108091 (eighth root of E_4), A341871 - A341875, A377973, A377975, A377976, A377977.
%K A377974 sign,easy
%O A377974 0,3
%A A377974 _Peter Bala_, Nov 13 2024